It is similar with division and multiplication. Just as people sometimes want to form sets of the same size into one larger set, they sometimes want to break up a large set into equal-sized pieces. If you think of 15 as 5×3, then when you divide 15 by 3, you are again left with 5. Thus division by 3 undoes implicit multiplication by 3 and leaves you with the original amount. It is the same no matter what amount you start with: 5×3÷3=5; 9×3÷3=9; 743×3÷3=743. More formally, dividing by 3 is the inverse of multiplying by 3.

Two interpretations of division deserve particular mention here. If I have 20 cookies, and want to sort them into 5 bags, how many go in each bag? This is the so-called sharing model of division because I know in how many ways the cookies are to be shared. I can find the answer by picturing the 20 cookies arranged in 5 groups of 4 cookies, which will be the contents of 1 bag. If the cookies originally came out of 5 bags of 4 each, when I put them back into those bags, I will again have 4 in each. Thus, division by 5 undoes multiplication by 5, or division by 5 is the inverse of multiplication by 5. The picture below shows the sharing model for this situation.

To think about 20÷5, you could also use the measurement model: If I have 20 cookies that are to be packaged in bags of 5 each, how many bags will I get? In the sharing model (also called the partitioning model or partitive division), you know the number of groups and seek the number in a group. In the measurement model (also called quotative division), you know the size of the groups and seek the number of groups. The circled numbers in the figures above and below illustrate a crucial difference between the two models: the order in which the cookies are placed in bags. In the sharing

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